and present

Introduction to Introduction to Derivatives

Pulkit Singhal & Kush Shah

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F I N G Y A A N S E S S I O N 1

What are derivatives?What are derivatives?

Derivatives are financial instruments h i d d whose prices depend on, or are

derived from, the prices of other assetsassets.

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What are the assets on which their prices depend?

Underlying is financial in natureUnderlying is financial in natureStock pricesCredit ratingInterest rateste est atesExchange rates

Also..ElectricityyThe weatherInsuranceCattle prices

In sum, if you can price the underlying asset, there can always be a derivative on it.

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Why are derivatives useful?Why are derivatives useful?

Hedging S l iSpeculationArbitrage profit-makingB l h t hBalance sheet changes

To change the nature of a liabilityTo change the nature of an To change the nature of an investment/assets

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The various kinds of derivativesThe various kinds of derivatives

There are three principal classes of derivative securities:

OptionsFutures and ForwardsSwaps

In addition, it is possible to have options on futures, futures on options, swaptions Infinite complexity

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swaptions.. Infinite complexity.

Forward ContractsForward Contracts

A b ll An agreement to buy or sell an asset at a certain time in the future for a certain pricepNo daily settlement. When the contract expires, one party buys the asset for the agreed price from the asset for the agreed price from the other party.The contract is an over-the-counter (OTC) agreement between 2 institutions.

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Profit from Forward PositionsProfit from Forward Positions

ProfitProfit

Price of Underlyingat Maturity

Price of Underlyingat Maturity

Long position Short position

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Working of a forward contractWorking of a forward contract

Exchange Rate(INR/USD)

Bid Ask

Spot 39.94 40.11

1-month 39.78 40.02

3 th 39 55 39 943-month 39.55 39.94

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Working of a forward contractWorking of a forward contract

Suppose Reliance has to make a Suppose Reliance has to make a payment of $25 M to Total 3 months from now and wish to lock in an exchange rate.They would enter a 3-month forward ycontract to buy $ at the 3-month forward exchange rate.What happens if the exchange rate 3-months from now is different from 39 94?

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39.94?

Futures ContractsFutures Contracts

An agreement to buy or sell an asset at a certain time in the future f t i ifor a certain price.They are exchange-traded, and h t d di d t thence, are standardized contracts.Important futures exchanges: CBOT, CME, NYMEX etc. In India: BSE, NSE, MCX, NCDEX.

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Examples of Futures ContractsExamples of Futures Contracts

Agreement to:

Buy 1000 shares of Reliance at Rs. 2500 in May (BSE) y ( )Sell $1 million at 1.5000 US$/£ in april (CME)in april (CME)

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Futures ArbitrageFutures Arbitrage

Suppose that:Suppose that:

The spot price of Reliance is p pRs. 1500The quoted 1-year futures price is Rs 2000price is Rs. 20001-year interest rate is 10%

Is there an arbitrage opportunity?

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Futures ArbitrageFutures Arbitrage

Solution:Solution:Borrow Rs. 1500 at 10%Go long spotg pShort futures

End of year profit= 2000 – 1500*(1+10%)

What if the futures price is Rs 1600?

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is Rs. 1600?

Non-arbitrage futures priceIf the spot price is S and futures price is F for a If the spot price is S and futures price is F for a

contract deliverable in T years, then

F = S (1+r )T

If F > S (1+r )T, go short on futures and long on spot and vice versa.

What is the value of the futures contract?

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contract?

Daily settlement and margins

M i i h k t bl Margin is cash or marketable securities deposited by an investor with the brokerwith the brokerMarking to market: Balance in the margin account is adjusted to reflect margin account is adjusted to reflect daily settlementMargins guard against defaultMargins guard against defaultHow are margins set?

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Forward Contracts vs Futures Contracts

P i t t t b t 2 ti E h t d d

FORWARDS FUTURES

Private contract between 2 parties Exchange traded

Non-standard contract Standard contract

Usually 1 specified delivery date Range of delivery dates

Settled at maturity Settled daily

Delivery or final cashsettlement usually occurs

Contract usually closed outprior to maturity

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OptionsAn option is a security that gives the holder the An option is a security that gives the holder the right but not the obligation to buy or sell a security for a specified price at a specified date.

Basic classification of options:Basic classification of options:Call options/Put optionsAmerican options/European options

How are options different from f t /f d ?

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futures/forwards?

Intrinsic and Time ValueOption premium = Intrinsic Value + Time ValueOption premium Intrinsic Value + Time Value

Intrinsic value: payoff if option is exercised immediately always greater than or equal to zero immediately, always greater than or equal to zero.

Usually the price of an option in the marketplace will be greater than its intrinsic value The will be greater than its intrinsic value. The difference between the market value of an option and its intrinsic value is called the time value of an optionoption.

What are in-the-money, out-of-the-money and at the money options?

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at-the-money options?

Long Call

Profit from buying European call option: option price = Profit from buying European call option: option price = $5, strike price = $100, option life = 2 months

$30

20

Profit ($)

20

1070 80 90 100

Terminalt k i ($)

0-5

70 80 90 100

110 120 130

stock price ($)

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Short Call

Profit from writing a European call option: option price = Profit from writing a European call option: option price = $5, strike price = $100

Profit ($)

05 110 120 130

Profit ($)

-1070 80 90 100 Terminal

stock price ($)

-30

-20

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30

Long Put g

Profit from buying an European put option: option price Profit from buying an European put option: option price = $7, strike price = $70

30 Profit ($)30

20

Profit ($)

10 Terminalstock price ($)

0-7

70605040 80 90 100

stock price ($)

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Short Put

Profit from writing an European put option: option price Profit from writing an European put option: option price = $7, strike price = $70

Profit ($)70

605040

Profit ($)Terminal

stock price ($)

-10

070 80 90 100

-30

-20

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-30

Effect of Variables on Option pPricing

c p C PVariableS0 + + ––0XT

+ ++? ? + +– – +

σrD

+ + + ++ – + –+ +D – + – +

• What is the relation between price of an American option and a European option?

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option and a European option?

Put-Call Parity; No Dividends Put Call Parity; No Dividends

Consider the following 2 portfolios:Portfolio A: European call on a stock + PV of the strike price in cashthe strike price in cashPortfolio C: European put on the stock + the stock

Both are worth Max (ST , X) at the maturity of the optionsThey must therefore be worth the same todayy y

This means that

c + Xe -rT = p + S0

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p 0

Valuation using Black Scholes equationc S N d X e N drT= − −( ) ( )c S N d X e N dp X e N d S N drT

=

= − − −−

0 1 2

2 0 1

( ) ( )( ) ( )

d S X r TT

=+ +

10

2 2where ln( / ) ( / )σσ T

d S X r T d T=+ −

= −02 2ln( / ) ( / )

σ

σσd

Td T= =2 1

σσ

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What is Risk-Neutral Valuation?

1. Assume that the expected return from an asset is the return from an asset is the risk-free rate

2 Calculate the expected 2. Calculate the expected payoff from the derivative

3 Discount at the risk-free 3. Discount at the risk-free rate

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Naked and Covered PositionsNaked and Covered Positions

Naked positionNaked positionTake no action

Covered positionBuy 100,000 shares today

Both strategies leave the bank exposed to significant risk. H ?

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How?

Stop-Loss StrategyStop Loss Strategy

This involves:Buying 100,000 shares as soon

i h $50as price reaches $50Selling 100,000 shares as soon as price falls below $50as price falls below $50

Wh t i th bl ith What is the problem with this strategy?

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The Greeks: Delta The Greeks: Delta

Delta (Δ) is the rate of change of the Delta (Δ) is the rate of change of the option price with respect to the underlying

OptionOptionprice

Slope = Δ

A

BSlope Δ

Stock price

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A Stock price

Delta…Delta = sensitivity of an option's theoretical y pvalue to a change in the price of the underlying contract.

delta = change in the option pricechange in the stock pricechange in the stock price

What is the range of deltas for calls and gputs?

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Why is delta also called the hedge ratio?

ThetaTheta

h ( ) f d ( f l fTheta (Θ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of timevalue with respect to the passage of time

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GammaGamma

Gamma (Γ) is the rate of change of delta (Δ) with respect to the price of the underlying assetThe Gamma of a portfolio of derivatives on an underlying asset is the rate of change of the portfolio's delta with change of the portfolio s delta with respect to the price of the underlying asset. If gamma is large, delta is highly

iti t th i f th d l i sensitive to the price of the underlying asset.

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VegaVega

Vega (ν) is the rate of change of the value of a derivatives portfolio with respect to volatility

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Managing Delta, Gamma, & Vega

D lt Δ b h d b Delta, Δ, can be changed by taking a position in the

d lunderlying assetTo adjust gamma, Γ, and vega, ν,j g , , g , ,it is necessary to take a position in an option or other derivativein an option or other derivative

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RhoRho

Rh i th t f h f Rho is the rate of change of the value of a derivative with respect to the interest raterespect to the interest rate

For currency options there are 2 rhos2 rhos

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Hedging in PracticeHedging in Practice

Traders usually ensure that their portfolios are delta-neutral at least portfolios are delta neutral at least once a dayWhenever the opportunity arises Whenever the opportunity arises, they improve gamma and vegaAs portfolio becomes larger hedging As portfolio becomes larger hedging becomes less expensive

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Exotic OptionsBermudan option - non-standard American option in which Bermudan option - non-standard American option in which early exercise is limited to certain dates during the life of the option. Also referred to as "hybrid-style" exercise.

Forward start option is an option that is paid for now, but does not begin until some later date.

Compound option is an option on an option. Compound options have two strike prices and two expiration dates. For example, a call on a call is purchased. At some specified p , p pdate in the future, a person will have the right but not the obligation of purchasing a call option.

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Exotic OptionsChooser option also called an "as you like it" option allows Chooser option, also called an as you like it option, allows the holder to choose after a specified period of time whether the option is a call or a put.

Barrier option is an option in which the payoff depends on whether the underlying asset's price reaches a certain level during the life of the option.

Up-and-out option becomes worthless once the underlying asset price Up and out option becomes worthless once the underlying asset price reaches a specified boundary price. Up-and-in option requires the underlying asset price to reach the boundary price before the option can be activated.

Rainbow option is an option involving two or more risky assets.

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Exotic Options

Lookback option - payoffs depend on the maximum or minimum the stock price reaches over the life of the option.

Asian option (average price option) payoff depends on the Asian option (average price option) - payoff depends on the average price of the asset (not the stock price itself) over a specified amount of time during the life of the option.

Spread option - strike price is the spread between two underlying assets. For example, crack spreads on the spread between the price of crude and its by-products.

Basket option - payoff depends upon a portfolio of assets.

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Terms to watch out for

V l tilit t di l ti t diVolatility trading, correlation tradingDelta hedging, Gamma HedgingLong, short, spreadBasis riskLibor, yield curve

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Butterfly Spread Using CallsButterfly Spread: buying a call option with a Butterfly Spread: buying a call option with a relative low strike price, K1, buying a call option with a relative high strike price. K3, and selling two call options with a strike price halfway in between Kbetween, K2.Stock price Range

Payoff from First L C ll

Payoff from Second L C ll

Payoff from Short Calls

Total Payoff

Long Call Option

Long Call Option

ST ≥ K3 ST - K1 ST - K3 -2(ST - K2) 0K2 < ST < K3K2 < ST < K3ST ≤ K1

ST - K1ST - K10

000

-2(ST - K2) 00

K3 - STST - K10

Butterfly Spread Using CallsExample: Call option prices on a $61 stock are: $10 for a $55 Example: Call option prices on a $61 stock are: $10 for a $55 strike, $7 for a $60 strike, and $5 for a $65 strike. The investor could create a butterfly spread by buying one call with $55 strike price, buying a call with a $65 strike price, and selling two calls with a $60 strike pricewith a $60 strike price.

Stock price Range

Payoff from First Long Call

Payoff from Second Long Call

Payoff from Short Calls

Total Payoff

Long Call Option

Long Call Option

ST ≥ $65$60 < S

ST - $55S $55

ST - $650

-2(ST - $60) 2(S $60)

0$65 S$60 < ST

<$65$55 < ST<$60

ST - $55ST - $550

000

-2(ST - $60)00

$65 - STST -$550

<$60ST ≤ $55

Butterfly Spread Using Callsy p g

Profit

K1 K3 STK21 3 ST2

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Butterfly Spread Using Putsy p g

Profit

K1 K3 STK21 3 ST2

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